Nimber

In mathematics, nimbers, also called Grundy numbers, are important ideas used in a branch called combinatorial game theory. They help describe the values of heaps in a fun game called Nim. Nimbers are similar to ordinal numbers, but they have their own special ways of adding and multiplying, known as nimber addition and nimber multiplication.

Thanks to something called the Sprague–Grundy theorem, nimbers are useful in many types of games where players take turns. These games are called impartial games. Nimbers can even show up in games where players have different powers, like Domineering.

Nimber addition and multiplication follow certain rules. A special rule called the minimum excludant is used with sets of nimbers to find new values. These ideas help game theorists understand complex strategies and solve puzzles.

Definition

Nimbers are special numbers used in a game called Nim and in the study of combinatorial games. They were introduced by John Conway. Nimbers are different from regular numbers because they have unique rules for addition and multiplication. Even though nimbers can be matched with natural numbers, they work differently from everyday math.

Nimbers are written using a star, like ⁎0, ⁎1, ⁎2, and so on, to show their special role in game theory.

Uses

Nim

Main article: Nim

Nim is a game for two players. They take turns taking objects from different groups, or heaps. The player who takes the last object wins. The nimber of a heap is just the number of objects in it. Players can use nim addition to find a way to win.

Cram

Main article: Cram (game)

Cram is a game played on a rectangle board. Players take turns placing dominoes either across or up and down. The first player who cannot make a move loses. Like Nim, Cram is an impartial game and can have a nimber value. For example, boards that are even-sized by even-sized will have a nimber of 0.

Northcott's game

In Northcott's game, players move pegs up or down a column but cannot pass each other’s peg. Several columns make the game more complex. The player who cannot make a move loses. The spaces between pegs work like Nim heaps, and players can use Nim strategies to win.

Hackenbush

Main article: Hackenbush

Hackenbush is a game made by mathematician John Conway. Players take turns removing colored lines connected to a ground line. In its impartial version, either player can cut any branch, and any parts that depend on that branch fall off. Each place where a line connects to the ground can be thought of as a Nim heap with a nimber value.

Addition

Nimber addition, also called nim-addition, helps us find the size of a single Nim heap that is the same as a group of Nim heaps together. It is a special way of adding numbers for Nim games.

Nimber addition has some nice properties: it is associative (we can change the order of addition without changing the result) and commutative (the order of the numbers does not matter). The number 0 is the identity element, meaning adding 0 does not change the nimber. Every nimber is its own opposite, so adding a nimber to itself gives 0.

Multiplication

Nimber multiplication, also called nim-multiplication, has special rules that are different from normal multiplication. One rule is that when you multiply a Fermat 2-power, like 2²ⁿ, by a smaller number, the result is the same as regular multiplication. Another rule says that the nimber square of a Fermat 2-power equals 3·2²ⁿ⁻¹ when using regular multiplication.

These rules help us understand how nimbers work in calculations. The set of all finite nimbers connects to mathematical structures called fields, similar to systems used in advanced algebra.

Addition and multiplication tables

These tables show addition and multiplication for the first 16 nimbers. Nimbers are special numbers used in a game called Nim. They follow special rules for adding and multiplying. These rules are different from the usual rules you learn in school. The number 16 is a power of two (2 raised to the power of 4), which makes it useful for showing these special math operations.